
In three dimensions there are two analogues of polar coordinates.
In cylindric coordinates,
$x$
and
$y$
are described by
$r$
and
$\theta $
exactly as in two dimensions, while the third dimension,
$z$
is treated as an ordinary coordinate.
$r$
then represents distance from the
$z$
axis.
In spherical coordinates, a general point is described by two angles and one radial variable, $\rho $ , which represents distance to the origin: ${\rho}^{2}={x}^{2}+{y}^{2}+{z}^{2}$ .
The two angular variables are related to longitude and latitude, but latitude is zero at the equator, and the variable $\varphi $ that we use is 0 on the $z$ axis (which means at the north pole).
We define $\varphi $ by $\mathrm{cos}\varphi =\frac{z}{\rho}$ , so that with $r$ defined as always here by ${r}^{2}={x}^{2}+{y}^{2}$ , we have $\mathrm{sin}\varphi =\frac{r}{\rho}$ .
The longitude angle $\theta $ is defined by $\mathrm{tan}\theta =\frac{y}{x}$ , exactly as in two dimensions. We therefore have $x=r\mathrm{cos}\theta =\rho \mathrm{sin}\varphi \mathrm{cos}\theta $ , and what is $y$ ?
Exercises:
3.12 Express the parameters of cylindric and spherical coordinates in terms of $x,y$ and $z$ .
3.13 Construct a spreadsheet converter which takes coordinates $x,y$ and $z$ and produces the three parameters of spherical coordinates; and vice versa. Verify that they work by substituting the result from one as input into the other.
